L(s) = 1 | + 1.83·5-s + (2.45 − 0.997i)7-s + 3.09·11-s + (2.40 + 4.16i)13-s + (−1.87 − 3.24i)17-s + (−2.71 + 4.70i)19-s + 7.95·23-s − 1.62·25-s + (0.325 − 0.563i)29-s + (−0.518 + 0.897i)31-s + (4.50 − 1.83i)35-s + (0.873 − 1.51i)37-s + (−2.52 − 4.36i)41-s + (−6.09 + 10.5i)43-s + (−2.30 − 3.99i)47-s + ⋯ |
L(s) = 1 | + 0.821·5-s + (0.926 − 0.376i)7-s + 0.933·11-s + (0.666 + 1.15i)13-s + (−0.453 − 0.786i)17-s + (−0.622 + 1.07i)19-s + 1.65·23-s − 0.325·25-s + (0.0604 − 0.104i)29-s + (−0.0930 + 0.161i)31-s + (0.760 − 0.309i)35-s + (0.143 − 0.248i)37-s + (−0.393 − 0.682i)41-s + (−0.929 + 1.61i)43-s + (−0.336 − 0.582i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.378191654\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.378191654\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.45 + 0.997i)T \) |
good | 5 | \( 1 - 1.83T + 5T^{2} \) |
| 11 | \( 1 - 3.09T + 11T^{2} \) |
| 13 | \( 1 + (-2.40 - 4.16i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.87 + 3.24i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.71 - 4.70i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 7.95T + 23T^{2} \) |
| 29 | \( 1 + (-0.325 + 0.563i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.518 - 0.897i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.873 + 1.51i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.52 + 4.36i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.09 - 10.5i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.30 + 3.99i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.55 + 7.88i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.89 - 5.02i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.40 - 4.16i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.23 + 12.5i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.00T + 71T^{2} \) |
| 73 | \( 1 + (1.81 + 3.14i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.17 - 12.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.83 - 6.63i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.76 + 9.99i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.04 - 1.80i)T + (-48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.378823395368841682704904766649, −8.831788935804797458979185683910, −7.970678175423560288647536941901, −6.84446556645185606096067086384, −6.41561449654117168843358466188, −5.29045588187921729558885384051, −4.48408821960058419522709291809, −3.58036407060895930676106025383, −2.06293422155306714044827582864, −1.32334158711364736331110035450,
1.17678365211628822069884006628, 2.18889301954744984220079693472, 3.35434166339659995964640928660, 4.55754514260688268350773729321, 5.34570926495156232868097372980, 6.17518835488188662528342775128, 6.89244544109446159758900376739, 8.048359241521953594824981648614, 8.738028399123264908779571804021, 9.273651638298987065519613838365